Symmetry Sectors in Chord Space and Relational Holography in the DSSYK

TL;DR

This paper explores how gauging symmetries in the boundary theory of the DSSYK model affects its bulk dual, analyzing symmetry sectors, partition functions, and holographic interpretations in the context of quantum gravity.

Contribution

It introduces a detailed analysis of symmetry sectors in the DSSYK model and connects them to bulk dual proposals, including the study of Euclidean wormholes and baby universe Hilbert spaces.

Findings

Euclidean wormholes are perturbatively stable.

Baby universe Hilbert space is non-trivial with matter.

Holographic dictionary matches geodesic lengths with complexity.

Abstract

In holography, gauging symmetries of the boundary theory leads to important modifications in the bulk. In this work, we study constraints to gauge symmetry sectors in the chord Hilbert space of the double-scaled SYK (DSSYK) with matter, and we connect them to different proposals of its bulk dual. These sectors include chord parity symmetry, corresponding to End-Of-The-World (ETW) branes and Euclidean wormholes in sine dilaton gravity; and relative time-translations in a doubled DSSYK model (resulting from a single DSSYK with an infinitely heavy matter chord) used in de Sitter holography. We define and evaluate partition functions and thermal correlation functions of the ETW brane and Euclidean wormhole systems in the boundary theory. We deduce the holographic dictionary by matching geodesic lengths in the bulk with the spread complexity of the parity-gauged DSSYK. The Euclidean wormholes of fixed size are perturbatively stable, and their baby universe Hilbert space is non-trivial only when matter is added. We conclude studying the constraints in the path integral of the doubled DSSYK. We derive the gauge invariant operator algebra of one of the DSSYKs dressed to the other one and discuss its holographic interpretation.